Pairs of tiles which admit finitely or countably infinitely many tilings (Q1071318)

Let \({\mathcal P}\) be a set of prototiles, i.e. a finite set of tiles. If a tiling exists such that every tile of it is congruent to a prototile from \({\mathcal P}\), we say that \({\mathcal P}\) admits a tiling. It is proved that for every \(k\geq 2\) and \(r\geq 1\) there exists a set of k prototiles which admits exactly r distinct tilings. The author works with the case of 2 prototiles and later easy derives the solution for every \(k>2\). It is also shown that there exist pairs of prototiles which admit countable infinity of different tilings.